Rotation in different reference frames

In this notebook we want to look at the basics of rotation in different reference frames. To do this we will look at vectors rotating in Euclidian (maybe change this name so it's more intuitive?) space, and in and Minkowski spacetime.

Recall that in Euclidian space on a 2 dimensional plane we have two spatial coordinates:

$$ \mathbf{x} = \left(\begin{array}{l}x\\y\end{array}\right) $$

(maybe change this name so it's more intuitive?)

First we need to import the relevent functions

%load_ext autoreload
%autoreload 2
import modules.rotation as rt

%%capture
animation = rt.animate_plot_1()

animation

If we consider two vectors moving through Euclidian space in this manner there are two things we can change. Firstly, we can change the angle of the first vector, keeping the angle beween the two vectors constant, or we could change the angle between the two vectors, keeping the podsition of one vector constant. (again I think this could be written better)

%%capture
animation_1, animation_2 = rt.animate_2_euclidian_vedctors()

animation_1

Potentially need to discuss what is meant by the dot product.

animation_2

Working in Minkowski Space time

In Minkowski spacetime our equivilent to angle ($\theta$) is velocity ($u$). As explored in the lorentz notebook, if we change the velocity of an object or observer, time dilation will occur and instead of tracing out a circle, Hyperbolae are generated for a constant "distance" in Minkowski spacetime.

%%capture
from modules.lorentz import animation_with_hyperbolae as lorentz_example
animation = lorentz_example()

animation

These hyperbolae can be described by finding where $(ct)^2-x^2$ is constant. This can help us now if we consider two objects moving through Minkowski spacetime, each with a different velocity. As before, we can consider what happens when the diffence in velocities stays constant and what happens when one velocity stays constant and the other one changes.

%%capture
animation = rt.Minkowski_2_vectors_animate(vec1 = [0,9], vec2 = [3,7])

animation